publications

Journal articles

2023

1. NA

   Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives Miśkiewicz, Michał, Petraszczuk, Bogdan, and Strzelecki, Paweł Nonlinear Analysis, 232, 21pp, 2023 [PDF]

2019

1. WiadMat

   Bogdan Bojarski (1931–2018) Strzelecki, Paweł Wiadomości Matematyczne 55, 231–251, 2019 [PDF]

2018

1. CAG

   Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies Kolasiński, Sławomir, Strzelecki, Paweł, and von der Mosel, Heiko Comm. Anal. Geom. 26, 1251–1316, 2018 [arXiv] [HTML] [PDF]

2017

1. ACV

   The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data Mazowiecka, Katarzyna, and Strzelecki, Paweł Adv. Calc. Var. 10, 303–314, 2017 [HTML] [PDF]
2. EMSS

   Invitation to H-systems in higher dimensions: known results, new facts, and related open problems Schikorra, Armin, and Strzelecki, Paweł EMS Surv. Math. Sci. 4, 21–42, 2017 [arXiv] [HTML] [PDF]

2016

1. NoDEA

   A conditional regularity result for p-harmonic flows Kazaniecki, Krystian, Łasica, Michał, Mazowiecka, Katarzyna Ewa, and Strzelecki, Paweł NoDEA Nonlinear Differential Equations Appl. 23, Art. 9, 13, 2016 [HTML] [PDF]

2013

1. PhysRep

   Menger curvature as a knot energy Strzelecki, Paweł, and von der Mosel, Heiko Phys. Rep. 530, 257–290, 2013 [Abs] [HTML] [PDF]

   Motivated by the suggestions of Gonzalez and Maddocks, and Banavar et al. to use geometrically defined curvature energies to model self-avoidance phenomena for strands and sheets we give a self-contained account, aimed at non-experts, on the state of art of the mathematics behind these energies. The basic building block, serving as a multipoint potential, is the circumradius of three points on a curve. The energies we study are defined as averages of negative powers of that radius over all possible triples of points along the curve (or via a mixture of averaging and maximization). For a suitable range of exponents, above the scale invariant case, we establish self-avoidance and regularizing effects and discuss various applications in geometric knot theory, as well as generalizations to surfaces and higher-dimensional submanifolds.

2. Topo&Appl

   On some knot energies involving Menger curvature Strzelecki, Paweł, Szumańska, Marta, and von der Mosel, Heiko Topology Appl. 160, 1507–1529, 2013 [HTML] [PDF]
3. JGA

   Tangent-point repulsive potentials for a class of non-smooth m-dimensional sets in Rⁿ. Part I: Smoothing and self-avoidance effects Strzelecki, Paweł, and von der Mosel, Heiko J. Geom. Anal. 23, 1085–1139, 2013 [HTML] [PDF]
4. GAFA

   Characterizing W^2,p submanifolds by p-integrability of global curvatures Kolasiński, Sławomir, Strzelecki, Paweł, and von der Mosel, Heiko Geom. Funct. Anal. 23, 937–984, 2013 [HTML] [PDF]
5. Studia

   Weak compactness of solutions for fourth order elliptic systems with critical growth Goldstein, Paweł, Strzelecki, Paweł, and Zatorska-Goldstein, Anna Studia Math. 214, 137–156, 2013 [HTML] [PDF]

2012

1. JKTR

   Tangent-point self-avoidance energies for curves Strzelecki, Paweł, and von der Mosel, Heiko J. Knot Theory Ramifications 21, 1250044, 28, 2012 [HTML] [PDF]

2011

1. AdvMath

   Integral Menger curvature for surfaces Strzelecki, Paweł, and von der Mosel, Heiko Adv. Math. 226, 2233–2304, 2011 [HTML] [PDF]

2010

1. AnnSNSPisa

   Regularizing and self-avoidance effects of integral Menger curvature Strzelecki, Pawel, Szumańska, Marta, and von der Mosel, Heiko Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9, 145–187, 2010 [PDF]

2009

1. Studia

   On polyharmonic maps into spheres in the critical dimension Goldstein, Paweł, Strzelecki, Paweł, and Zatorska-Goldstein, Anna Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1387–1405, 2009 [HTML] [PDF]
2. AASF

   A geometric curvature double integral of Menger type for space curves Strzelecki, Paweł, Szumańska, Marta, and von der Mosel, Heiko Ann. Acad. Sci. Fenn. Math. 34, 195–214, 2009 [HTML]

2008

1. Manuscripta

   A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two Hajłasz, Piotr, Strzelecki, Paweł, and Zhong, Xiao Manuscripta Math. 127, 121–135, 2008 [Abs] [HTML] [PDF]

   We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far.

2. ACV

   On a nonlinear fourth order elliptic system with critical growth in first order derivatives Strzelecki, Paweł, and Zatorska-Goldstein, Anna Adv. Calc. Var. 1, 205–222, 2008 [HTML] [PDF]

2007

1. MathZ

   On rectifiable curves with L^p-bounds on global curvature: self-avoidance, regularity, and minimizing knots Strzelecki, Paweł, and von der Mosel, Heiko Math. Z. 257, 107–130, 2007 [HTML] [PDF]

2006

1. BullLMS

   Gagliardo-Nirenberg inequalities with a BMO term Strzelecki, P. Bull. London Math. Soc. 38, 294–300, 2006 [HTML] [PDF]
2. CalcVar

   Global curvature for surfaces and area minimization under a thickness constraint Strzelecki, Paweł, and von der Mosel, Heiko Calc. Var. Partial Differential Equations 25, 431–467, 2006 [Abs] [HTML] [PDF]

   Motivated by previous work on elastic rods with self-contact, involving the concept of the global radius of curvature for curves (as defined by Gonzalez and Maddocks), we define the global radius of curvature ∆[X] for a wide class of continuous parametric surfaces for which the tangent plane exists on a dense set of parameters. It turns out that in this class of surfaces a positive lower bound ∆[X] ≥θ> 0 provides, naively speaking, the surface with a thickness of magnitude θ; it serves as an excluded volume constraint for X, prevents self-intersections, and implies that the image of X is an embedded C¹-manifold with a Lipschitz continuous normal. We also obtain a convergence and a compactness result for such thick surfaces, and show one possible application to variational problems for embedded objects: the existence of ideal surfaces of fixed genus in each isotopy class. The proofs are based on a mixture of elementary topological, geometric and analytic arguments, combined with a notion of the reach of a set, introduced by Federer in 1959.

3. Monthly

   The Poincaré conjecture? Strzelecki, Paweł Amer. Math. Monthly 113, 75–78, 2006 [HTML]

2005

1. Manuscripta

   Sard’s theorem for mappings in Hölder and Sobolev spaces Bojarski, Bogdan, Hajłasz, Piotr, and Strzelecki, Paweł Manuscripta Math. 118, 383–397, 2005 [Abs] [HTML] [PDF]

   We prove various generalizations of classical Sard’s theorem to mappings f:M^m—> Nⁿ between manifolds in Hölder and Sobolev classes. It turns out that if f∈C^k,λ(M^m,Nⁿ), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f^-1(y). The classical theorem of Sard holds true for f∈C^k with sufficiently large k, i.e., k>max (m-n, 0); our estimates contain Sard’s theorem (and improvements due to Dubowicki and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f^-1(y).

2. CommPDE

   A sharp nonlinear Gagliardo-Nirenberg-type estimate and applications to the regularity of elliptic systems Rivière, Tristan, and Strzelecki, Paweł Comm. Partial Differential Equations 30, 589–604, 2005 [HTML] [PDF]
3. Monthly

   How to measure volume with a thread Hajłasz, Piotr, and Strzelecki, Paweł Amer. Math. Monthly 112, 176–179, 2005 [HTML] [PDF]

2004

1. Duke

   A compactness theorem for weak solutions of higher-dimensional H-systems Strzelecki, Paweł, and Zatorska-Goldstein, Anna Duke Math. J. 121, 269–284, 2004 [HTML] [PDF]

2003

1. Manuscripta

   Regularity of generalized sphere valued p-harmonic maps with small mean oscillations Strzelecki, Paweł Manuscripta Math. 112, 473–487, 2003 [HTML] [PDF]
2. CalcVar

   On biharmonic maps and their generalizations Strzelecki, Paweł Calc. Var. Partial Differential Equations 18, 401–432, 2003 [Abs] [HTML] [PDF]

   We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hoelder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.

3. CalcVar

   A new proof of regularity of weak solutions of the H-surface equation Strzelecki, Paweł Calc. Var. Partial Differential Equations 16, 227–242, 2003 [HTML] [PDF]

2002

1. IUMJ

   Improved C^k,λ approximation of higher order Sobolev functions in norm and capacity Bojarski, Bogdan, Hajłasz, Piotr, and Strzelecki, Paweł Indiana Univ. Math. J. 51, 507–540, 2002 [HTML] [PDF]

2001

1. IUMJ

   Hardy space estimates for higher-order differential operators Strzelecki, Paweł Indiana Univ. Math. J. 50, 1447–1461, 2001 [Abs] [HTML] [PDF]

   We generalize a theorem of Coifman, Lions, Meyer and Semmes to higher order differential operators (also with variable coefficients). Moreover, we present a direct, elementary proof of a duality inequality which is a special case of the duality of Hardy space and BMO. The tools used in this proof come from standard Sobolev space theory; no knowledge of Hardy spaces is required.

1998

1. MathAnn

   Subelliptic p-harmonic maps into spheres and the ghost of Hardy spaces Hajłasz, Piotr, and Strzelecki, Paweł Math. Ann. 312, 341–362, 1998 [HTML] [PDF]

1996

1. Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions Strzelecki, Paweł Colloq. Math. 70, 271–289, 1996 [HTML] [PDF]

1994

1. Manuscripta

   Regularity of p-harmonic maps from the p-dimensional ball into a sphere Strzelecki, Paweł Manuscripta Math. 82, 407–415, 1994 [HTML] [PDF]

1993

1. On the differentiability of solutions of quasilinear elliptic equations Hajłasz, Piotr, and Strzelecki, Paweł Colloq. Math. 64, 287–291, 1993 [HTML] [PDF]

1992

1. Pointwise differentiability properties of solutions of quasilinear parabolic equations Strzelecki, Paweł Hokkaido Math. J. 21, 543–567, 1992 [HTML]
2. AASF

   Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients Strzelecki, Paweł Ann. Acad. Sci. Fenn. Ser. A I Math. 17, 171–180, 1992 [HTML]

Proceedings papers and book chapters

1. Geometric curvature energies: facts, trends, and open problems Strzelecki, Paweł, and von der Mosel, Heiko In: New directions in geometric and applied knot theory. De Gruyter, Berlin, 2018
2. How averaged Menger curvatures control regularity and topology of curves and surfaces Strzelecki, Paweł, and von der Mosel, Heiko Journal of Physics: Conference Series 544, 012018, 2014 [Abs]

   We demonstrate how integrated Menger curvatures can replace the nonsmooth ropelength functional to control regularity and topology of knotted curves and surfaces.

3. Convergence of weak solutions of the equation of hypersurface with prescribed mean curvature Zatorska-Goldstein, Anna, and Strzelecki, Paweł In: Mathematical approach to nonlinear phenomena: modelling, analysis and simulations. Gakko̵tosho, Tokyo, 2005
4. On a mathematical model for thick surfaces Strzelecki, Paweł, and von der Mosel, Heiko In: Physical and numerical models in knot theory. World Sci. Publ., Singapore, 2005
5. Quasilinear elliptic systems of Ginsburg-Landau type Strzelecki, P. In: Free boundary problems, theory and applications (Zakopane, 1995). Longman, Harlow, 1996
6. Stationary p-harmonic maps into spheres Strzelecki, Paweł In: Singularities and differential equations (Warsaw, 1993). Polish Acad. Sci. Inst. Math., Warsaw, 1996
7. Regularity of p-harmonic functions on a Riemann surface Strzelecki, Paweł In: Proceedings of the Fourth Finnish-Polish Summer School in Complex Analysis at Jyväskylä (1992). Univ. Jyväskylä, Jyväskylä, 1993